On the le-semigroups whose semigroup of bi-ideal elements is a normal band
It is well known that the semigroup B(S) of all bi-ideal elements of an le-semigroup S is a band if and only if S is both regular and intra-regular. Here we show that B(S) is a band if and only if it is a normal band and give a complete characterization of the le-semigroups S for which the associate...
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Інститут прикладної математики і механіки НАН України
2015
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irk-123456789-1551482019-06-17T01:27:07Z On the le-semigroups whose semigroup of bi-ideal elements is a normal band Bhuniya, A.K. Kumbhakar, M. It is well known that the semigroup B(S) of all bi-ideal elements of an le-semigroup S is a band if and only if S is both regular and intra-regular. Here we show that B(S) is a band if and only if it is a normal band and give a complete characterization of the le-semigroups S for which the associated semigroup B(S) is in each of the seven nontrivial subvarieties of normal bands. We also show that the set Bm(S) of all minimal bi-ideal elements of S forms a rectangular band and that Bm(S) is a bi-ideal of the semigroup B(S). 2015 Article On the le-semigroups whose semigroup of bi-ideal elements is a normal band / A.K. Bhuniya, M. Kumbhakar // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 171-181. — Бібліогр.: 24 назв. — англ. 1726-3255 2010 MSC:06F05. http://dspace.nbuv.gov.ua/handle/123456789/155148 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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It is well known that the semigroup B(S) of all bi-ideal elements of an le-semigroup S is a band if and only if S is both regular and intra-regular. Here we show that B(S) is a band if and only if it is a normal band and give a complete characterization of the le-semigroups S for which the associated semigroup B(S) is in each of the seven nontrivial subvarieties of normal bands. We also show that the set Bm(S) of all minimal bi-ideal elements of S forms a rectangular band and that Bm(S) is a bi-ideal of the semigroup B(S). |
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Bhuniya, A.K. Kumbhakar, M. |
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Bhuniya, A.K. Kumbhakar, M. On the le-semigroups whose semigroup of bi-ideal elements is a normal band Algebra and Discrete Mathematics |
| author_facet |
Bhuniya, A.K. Kumbhakar, M. |
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Bhuniya, A.K. |
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On the le-semigroups whose semigroup of bi-ideal elements is a normal band |
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On the le-semigroups whose semigroup of bi-ideal elements is a normal band |
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On the le-semigroups whose semigroup of bi-ideal elements is a normal band |
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On the le-semigroups whose semigroup of bi-ideal elements is a normal band |
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On the le-semigroups whose semigroup of bi-ideal elements is a normal band |
| title_sort |
on the le-semigroups whose semigroup of bi-ideal elements is a normal band |
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Інститут прикладної математики і механіки НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/155148 |
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On the le-semigroups whose semigroup of bi-ideal elements is a normal band / A.K. Bhuniya, M. Kumbhakar // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 171-181. — Бібліогр.: 24 назв. — англ. |
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Algebra and Discrete Mathematics |
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2025-07-14T07:14:19Z |
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1837605595085012992 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 2, pp. 171–181
© Journal “Algebra and Discrete Mathematics”
On the le-semigroups whose semigroup
of bi-ideal elements is a normal band
A. K. Bhuniya, M. Kumbhakar
Communicated by V. Mazorchuk
Abstract. It is well known that the semigroup B(S) of all
bi-ideal elements of an le-semigroup S is a band if and only if S is
both regular and intra-regular. Here we show that B(S) is a band if
and only if it is a normal band and give a complete characterization
of the le-semigroups S for which the associated semigroup B(S) is
in each of the seven nontrivial subvarieties of normal bands. We
also show that the set Bm(S) of all minimal bi-ideal elements of
S forms a rectangular band and that Bm(S) is a bi-ideal of the
semigroup B(S).
1. Introduction
In the ideal theory of commutative rings, it was observed by W. Krull
[15] that several results do not depend on the fact that the ideals are
composed of elements. The same is true for the ideal theory of semigroups
also. Consequently, these results can be formulated in a more general
setting of lattice-ordered semigroups where an element represents an
ideal of the ring or semigroup as an undivided entity. There are series of
articles dealing with lattice-ordered semigroups, generalizing theorems
from commutative ideal theory [1], [3], [5] and from the ideal theory of
semigroups [12], [13], [21], [22]. Presently, lattice ordered semigroups are
providing us a general setting not only for ‘abstract ideal theory’, but also
2010 MSC: 06F05.
Key words and phrases: bi-ideal elements, duo; intra-regular, lattice-ordered
semigroup, locally testable, normal band, regular.
172 On the le-semigroups
for order-preserving transformations of a finite chain, power semigroups
of an arbitrary semigroup, and for many other areas of algebra where the
objects form similar kinds of lattice-ordered semigroups.
In the present paper we study le-semigroups globally; our aim here
is to find out to what extent properties of the subsemigroup B(S) of
all bi-ideal elements of an le-semigroup S affect the structure of the le-
semigroup as a whole. In 1952, R.A. Good and D.R. Hughes [6] introduced
the notion of bi-ideals of a semigroup; these have been generalized again
and again to rings, semirings, ternary semirings, Γ-semigroups, etc [4],
[8]–[11], [14], [23]. It has also been proved that this notion is very useful
for characterizing different types of regularity of rings, semirings, and
semigroups [2], [16]– [19]. In [12], N. Kehayopulu defined bi-ideal elements
of an le-semigroup as a generalization of bi-ideals. Here we introduce the
notion of minimal bi-ideal elements and show that the product of any
two bi-ideal elements is a bi-ideal element, and that the product of any
two minimal bi-ideal elements is a minimal bi-ideal element. Thus the set
B(S) of all bi-ideal elements and the set Bm(S) of all minimal bi-ideal
elements are subsemigroups of S. It is well known that S is both regular
and intra-regular if and only if b2 = b for every bi-ideal element b of S,
equivalently B(S) is a band. Here we show that B(S) is a locally testable
semigroup and hence a normal band (since a band is locally testable if
and only if it is a normal band) if S is both regular and intra-regular.
The variety of normal bands has exactly eight subvarieties. Here we have
characterized the le-semigroups S such that B(S) is in each of these
subvarieties of normal bands.
This introduction is followed by preliminaries. In Section 3, we char-
acterize the le-semigroups S such that B(S) is in each of the subvarieties
of normal bands. In the last section, we show that the semigroup Bm(S)
of all minimal bi-ideal elements of S is a bi-ideal of the semigroup B(S)
whereas the set Lm(S) of all minimal left ideal elements is a left ideal of
the semigroup L(S) of all left ideal elements of S.
2. Preliminaries and foundations
An le-semigroup S is an algebra (S, ·, ∨, ∧, e) such that (S, ·) is a
semigroup, (S, ∨, ∧, e) is a lattice with a greatest element which is denoted
by e, and for all a, b, c ∈ S,
a(b ∨ c) = ab ∨ ac and (a ∨ b)c = ac ∨ bc.
A. K. Bhuniya, M. Kumbhakar 173
For different examples and relevance, both classical and modern, of
the le-semigroups we refer to [21]. Throughout the paper S will stand for
an le-semigroup (S, ·, ∨, ∧, e).
The usual order relation 6 on the set S is defined by: for a, b ∈ S
a 6 b if a ∨ b = b.
Since the multiplication is distributive over the lattice join, it follows that
the order 6 is compatible with the multiplication in S, that is, for all
a, b, c ∈ S,
a 6 b =⇒ ac 6 bc and ca 6 cb.
Let A be a nonempty subset of S. We denote (A] = {x ∈ S | x 6
a for some a ∈ A}. A nonempty subset L is called a left (right) ideal of
S if SL ⊆ L (LS ⊆ L) and (L] ⊆ L. A subset I is called an ideal if it is
both a left and a right ideal of S. For a ∈ S, the left ideal generated by a
is given by
(a]l = {x ∈ S | x 6 sa for some s ∈ S ∪ {1}}.
An element a ∈ S is called regular if a 6 aea; and intra-regular if
a 6 ea2e. If every element of S is regular (intra-regular) then the le-
semigroup S is defined to be regular (intra-regular). We also say that a is
(i) a subsemigroup element if a2 6 a;
(ii) a left ideal element if ea 6 a;
(iii) a right ideal element if ae 6 a;
(iv) a bi-ideal element if it is a subsemigroup element and aea 6 a.
From the above definitions it is evident that every left and right
ideal element is also a subsemigroup element. The definition of bi-ideal
elements that we have given here is a little bit different from that of
bi-ideal elements considered by Kehayopulu [12], Pasku and Petro [22].
According to these authors, a bi-ideal element b needs not satisfy b2 6 b,
i.e. needs not be a subsemigroup element, and is actually an abstraction
of the generalized bi-ideals (of a semigroup) and not of the bi-ideals.
Let a ∈ S. Then b = a ∨ a2 ∨ aea is the least bi-ideal element in S
such that a 6 b. We call a ∨ a2 ∨ aea the bi-ideal element generated by a,
and denote this by β(a). Thus a ∈ S is a bi-ideal element if and only if
β(a) = a.
Now we recall some notions of semigroups (without order). A semi-
group F is called regular if for every a ∈ F there is x ∈ F such
174 On the le-semigroups
that a = axa. By a band we mean a semigroup B such that b2 = b
for all b ∈ B. A band S is normal if for all a, b, c ∈ S, abca = acba. A
subsemigroup B of a semigroup F is called a bi-ideal of F if BFB ⊆ B.
In the diagram below, we use the following symbols to denote the
different subvarieties of normal bands.
Normal band N B abcd = acbd,
Rectangular band ReB aba = a,
Left normal band LN B abc = acb,
Right normal band RN B abc = bac,
Left zero band LZB ab = a,
Right zero band RZB ab = b,
Semilattice Sl ab = ba,
Trivial semigroup T a = b.
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A semigroup is called locally finite if every finitely generated subsemi-
group is finite. A locally testable semigroup [24] is a semigroup which is
locally finite and in which fSf is a semilattice for all idempotent f ∈ S.
Nambooripad [20] proved that a regular semigroup S is locally testable if
and only if fSf is a semilattice for all idempotent f ∈ S.
We refer the reader to [7] for the fundamentals of semigroup theory.
A. K. Bhuniya, M. Kumbhakar 175
3. Subsemigroup of all bi-ideal elements
We denote the set of all left, right, and bi-ideal elements of S by
L(S), R(S), and B(S), respectively. Then L(S), R(S), and B(S) are all
nonempty, since e is a left ideal, a right ideal, and a bi-ideal element of S.
Now for any two bi-ideal elements a and b of S, (ab)2 = (aba)b 6 ab and
abeab = (abea)b 6 ab, since a is a bi-ideal element of S, which shows that
the product of any two bi-ideal elements is a bi-ideal element. Thus B(S)
is a subsemigroup of S. Similarly both L(S) and R(S) are subsemigroups
of S.
Now we show that the regularity of an le-semigroup is equivalent to
the regularity of the semigroup B(S). This, we think, is well known. But
as we have seen the sufficient part nowhere, for the sake of completeness,
we include a proof.
Proposition 3.1. Let S be an le-semigroup. Then S is regular if and
only if the semigroup B(S) of all bi-ideal elements is regular.
Proof. First assume that S is regular and that b ∈ B(S). Since b is a
bi-ideal element, beb 6 b. On the other hand, b 6 beb by the regularity of
S. Thus we have b = beb which shows that b is a regular element in B(S),
since e is also a bi-ideal element of S.
Conversely, suppose that B(S) is a regular semigroup. Consider a ∈ S.
Then β(a) = a ∨ a2 ∨ aea ∈ B(S) and so there is b ∈ B(S) such that
a∨a2∨aea = (a∨a2∨aea)b(a∨a2∨aea) 6 (a∨a2∨aea)e(a∨a2∨aea) 6 aea.
This implies that a 6 aea. Thus S is a regular le-semigroup.
If S is a regular le-semigroup, then for every a ∈ S, a 6 aea implies
that a2 6 aaea 6 aea. Hence the bi-ideal element β(a) generated by
a reduces to the form β(a) = aea. Thus in a regular le-semigroup the
notions of bi-ideal elements as we have defined and that defined by N.
Kehayopulu [12] are the same. Therefore in a regular le-semigroup S,
an element b ∈ S is a bi-ideal element if and only if b = ca for some
right ideal element c and left ideal element a [12, Lemma 2]. This can be
reframed as:
Theorem 3.2. Let S be an le-semigroup. Then R(S)L(S) ⊆ B(S). If
moreover, S is a regular le-semigroup, then B(S) = R(S)L(S).
We also omit the proof of the following result, since this can be proved
easily:
176 On the le-semigroups
Proposition 3.3. Let S be a regular le-semigroup. Then R(S) and L(S)
are bands.
The following important result can be proved similarly to that in [13]
for the quasi-ideal elements.
Theorem 3.4. An le-semigroup S is both regular and intra-regular if
and only if B(S) is a band.
Now we show that B(S) is in fact a normal band if S is both a regular
and intra-regular le-semigroup.
Theorem 3.5. Let S be an le-semigroup. Then S is both regular and
intra-regular if and only if B(S) is a normal band.
Proof. Let a, b, c ∈ B(S). Then (bab)(bcb) = ba(bbcb) 6 ba(beb) 6 bab.
Similarly, (bab)(bcb) 6 bcb. Thus (bab)(bcb) 6 (bab) ∧ (bcb). Now let
u = (bab) ∧ (bcb). Then u 6 bab and u 6 bcb. Since S is both regular and
intra-regular, so B(S) is a band. Now ueu = (bab ∧ bcb)e(bab ∧ bcb) =
babebab ∧ babebcb ∧ bcbebab ∧ bcbebcb 6 bab ∧ babebcb ∧ bcbebab ∧ bcb 6
bab∧bcb = u shows that u ∈ B(S) which implies that u = u2 6 (bab)(bcb).
Thus (bab) ∧ (bcb) 6 (bab)(bcb) and hence (bab)(bcb) = (bab) ∧ (bcb).
Then bB(S)b = {bab | a ∈ B(S)} is a semilattice for every b ∈
B(S). Thus B(S) is a locally testable semigroup. Since a locally testable
semigroup is a band if and only if it is a normal band [24, Theorem 5], so
B(S) is a normal band.
The converse follows from the Theorem 3.4.
An ordered semigroup S is said to be left (right) duo if every left
(right) ideal of S is a right (left) ideal of S; and S is said to be duo if S
is both left and right duo.
Lemma 3.6. An le-semigroup S is left duo if and only if ae 6 ea for all
a ∈ S.
Proof. First assume that S is left duo and let a ∈ S. Then the left ideal
(a]l = {x ∈ S | x 6 sa for some s ∈ S} generated by a is a right ideal
also. Then ae ∈ (a]l implies that there is some s ∈ S such that ae 6 sa
and this implies that ae 6 ea.
Conversely let L be a left ideal of S and a ∈ L. Then for every s ∈ S,
as 6 ae 6 ea ∈ L implies that as ∈ L. Thus L is a right ideal of S and
hence S is left duo.
A. K. Bhuniya, M. Kumbhakar 177
Immediately we have:
Proposition 3.7. An le-semigroup S is duo if and only if ae = ea for
all a ∈ S.
Let S be a regular left duo le-semigroup. Then for every a ∈ S,
a 6 aea 6 (ae)aea 6 ea2ea shows that S is intra-regular. Hence B(S) is
a band. In fact we have:
Theorem 3.8. An le-semigroup S is regular left duo if and only if B(S)
is a left normal band.
Proof. First assume that S is regular left duo. Then B(S) is a band.
Let a, b, c ∈ B(S). Then abc = (abc)(abc) = aabcabc 6 a(ae)cabc 6
a(ea)cabc 6 acabc = aca(bc)(bc) 6 acab(cb)e 6 acabecb 6 acb. Similarly
acb 6 abc. Thus abc = acb and hence B(S) is a left normal band.
Conversely, assume that B(S) is a left normal band. Then S is regular.
Also for every a ∈ S, both ea and aea are bi-ideal elements of S, and
hence ae = (ae)(ae)(ae) = (aea)(ea)e = (aea)e(ea) [since B(S) is a
normal band] = (aeae2)a 6 ea which shows that S is left duo.
The left-right dual of this theorem is as follows:
Theorem 3.9. An le-semigroup S is regular right duo if and only if B(S)
is a right normal band.
A band is a semilattice if and only if it is both a left and a right
normal band. Hence it follows immediately that:
Theorem 3.10. An le-semigroup S is regular duo if and only if B(S) is
a semilattice.
Theorem 3.11. Let S be an le-semigroup. Then B(S) is a rectangular
band if and only if S is regular and eae = ebe for all a, b ∈ S.
Proof. First assume that B(S) is a rectangular band and that a, b ∈ S.
Since B(S) is a band, so S is regular and hence β(a) = aea and β(b) = beb.
Then β(a) = β(a)β(b)β(a) implies that a 6 aea = (aea)(beb)(aea) 6 ebe.
Then eae 6 e2be2 6 ebe. Similarly β(b) = β(b)β(a)β(b) implies that
ebe 6 eae. Thus eae = ebe for all a, b ∈ S.
Conversely let a ∈ S. Since S is regular, so a 6 aea 6 aeaea 6 aea2ea,
by the given condition. Thus a 6 ea2e, and hence S is intra-regular.
Therefore B(S) is a band, by Theorem 3.4. Now let a, b be two bi-ideal
elements of S. Since a is a bi-ideal element and S is already known to
be regular, then aea = a, and so a = aea = aeaea = aeabaea = aba; and
hence B(S) is a rectangular band.
178 On the le-semigroups
Theorem 3.12. Let S be an le-semigroup. Then B(S) is a left zero band
if and only if S is regular and ae 6 eb for all a, b ∈ S.
Proof. First assume that B(S) is a left zero band and a, b ∈ S. Since B(S)
is band, so S is regular and hence β(ae) = ae2ae and β(b) = beb. Then
β(ae) = β(ae)β(b) implies that ae 6 ae2ae = (ae2ae)(beb) 6 eb. Thus
ae 6 eb for all a, b ∈ S.
Conversely let a ∈ S. Since S is regular, so a 6 aea 6 aeaea 6 ae2a2a,
by the given condition. Thus a 6 ea2e, and hence S is intra-regular.
Therefore B(S) is a band, by Theorem 3.4. Now let a, b be two bi-ideal
elements of S. Since S is regular, a = aea, so that ab 6 ae = ae(ae) 6
ae2a 6 aea = a and a = aeaea 6 ae(ae) 6 ae(eab) 6 (aea)b = ab. Thus
a = ab and hence B(S) is a left zero band.
The left-right dual of this theorem is as follows:
Theorem 3.13. Let S be an le-semigroup. Then B(S) is a right zero
band if and only if S is regular and ea 6 be for all a, b ∈ S.
4. Subsemigroup of all minimal bi-ideal elements
In this section we introduce minimal bi-ideal elements and minimal
left ideal elements, and show that the set of all minimal bi-ideal elements
of S is a subsemigroup of B(S).
Definition 4.1. Let S be an le-semigroup. A bi-ideal element b is said
to be minimal if for every bi-ideal element a of S,
a 6 b implies that a = b.
Minimal left (right) ideal elements are defined similarly.
We denote the set of all minimal bi-ideal, left ideal, and right ideal
elements of S by Bm(S), Lm(S), and Rm(S), respectively.
Now we show that Bm(S) is a subsemigroup of B(S). For this consider
a, b ∈ Bm(S). Then ab is a bi-ideal element. To check the minimality, let
c be a bi-ideal element such that c 6 ab. Then ca and bc are bi-ideal
elements such that ca 6 aba 6 a. Then by minimality of a we have ca = a.
Similarly, bc = b. Then ab = cabc 6 cec 6 c and hence c = ab. Thus
ab ∈ Bm(S).
Similarly, it can be proved that both Lm(S) and Rm(S) are subsemi-
groups of B(S).
We also have:
A. K. Bhuniya, M. Kumbhakar 179
Theorem 4.2. If S is an le-semigroup then Bm(S) = Rm(S)Lm(S).
Proof. First consider a ∈ Rm(S) and c ∈ Lm(S), and denote b = ac.
Then b is a bi-ideal element, by Theorem 3.2. To show the minimality of
b, let p 6 b be a bi-ideal element of S. Then pe is a right ideal element of
S and pe 6 be = ace 6 ae 6 a implies by the minimality of a as a right
ideal element that pe = a. Similarly we have ep = c, since c is a minimal
left ideal element. Then p 6 b = ac = peep 6 p implies that p = b; and so
b becomes a minimal bi-ideal element. Thus Rm(S)Lm(S) ⊆ Bm(S).
Now consider b ∈ Bm(S). Then be and eb are a right ideal element and
a left ideal element, respectively. Let a 6 be be a right ideal element of S.
Then ab is a bi-ideal element of S such that ab 6 beb 6 b, and so ab = b,
since b is a minimal bi-ideal element. Then a 6 be = abe 6 ae 6 a implies
that a = be. Thus be is a minimal right ideal element of S. Similarly eb is a
minimal left ideal element of S. Then beeb is a bi-ideal element, by Theorem
3.2. Now beeb 6 b implies that b = beeb; and so b ∈ Rm(S)Lm(S). Thus
Bm(S) ⊆ Rm(S)Lm(S). Hence Bm(S) = Rm(S)Lm(S).
Theorem 4.3. a) Let S be an le-semigroup such that the set Lm(S) of
all minimal left ideal elements is non-empty. Then Lm(S) is a left ideal
of the semigroup L(S). Moreover, Lm(S) is a right zero band.
b) Let S be an le-semigroup such that the set Rm(S) of all minimal right
ideal elements is non-empty. Then Rm(S) is a right ideal of the semigroup
R(S). Moreover, Rm(S) is a left zero band.
Proof. a) Let l ∈ L(S) and a ∈ Lm(S). Then la is a left ideal element
such that la 6 ea 6 a. This implies that la = a, since a is a minimal
left ideal element. Hence la ∈ Lm(S) and so L(S)Lm(S) ⊆ Lm(S). Thus
Lm(S) is a left ideal of L(S).
Now la = a for every l ∈ L(S) and a ∈ Lm(S) implies that ab = b for
every a, b ∈ Lm(S); and hence Lm(S) is a right zero band.
b) Follows as the left-right dual of a).
Now we characterize the semigroup Bm(S) of all minimal bi-ideal
elements of S.
Theorem 4.4. Let S be an le-semigroup such that the set Bm(S) of all
minimal bi-ideal elements is non-empty. Then Bm(S) is a bi-ideal of the
semigroup B(S). Moreover, Bm(S) is a rectangular band.
Proof. We have already shown that Bm(S) is a subsemigroup of B(S).
Now consider a, c ∈ Bm(S) and b ∈ B(S). Then abc is a bi-ideal element
180 On the le-semigroups
of S. To show the minimality of abc, let d 6 abc be a bi-ideal element of S.
Then da is a bi-ideal element of S and da 6 abca 6 a implies that a = da,
since a is a minimal bi-ideal element. Similarly minimality of c implies
that c = cd. Then abc = dabcd 6 d and so d = abc which shows that abc
is a minimal bi-ideal element of S. Thus Bm(S)B(S)Bm(S) ⊆ Bm(S) and
hence Bm(S) is a bi-ideal of B(S).
If b ∈ Bm(S), then b is a subsemigroup element of S and so b2 6 b.
Now minimality of b implies that b2 = b. Thus Bm(S) is a band. Let
a, b ∈ Bm(S). Then aba is a bi-ideal element such that aba 6 aea 6 a
which implies that aba = a, since a is a minimal bi-ideal element of S.
Thus Bm(S) is a rectangular band.
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Contact information
A. K. Bhuniya,
M. Kumbhakar
Department of Mathematics, Visva-Bharati,
Santiniketan-731235, India
E-Mail(s): anjankbhuniya@gmail.com
Received by the editors: 14.07.2014
and in final form 18.05.2015.
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